What causes the impact on sound quality with bridging?
There are probably a number of factors which contribute to that, to varying degrees depending on the particular amplifier and speakers. Including:
1)Most amplifiers will have lower distortion when driving higher impedances (e.g., 8 ohms of more) than when driving lower impedances (e.g., 4 ohms or less). As I indicated earlier, in bridged mode the amp will see the speaker's impedance divided by 2.
I suspect that effect will be mitigated to some extent in the case of your particular amplifier, because it includes autoformers having 2 ohm taps (which I suspect you would find to be the best choice when bridging with a 4 ohm speaker), as well as 4 ohm taps (which I suspect you would find to be the best choice when bridging with an 8 ohm speaker).
2)As you probably realize, speakers commonly have impedances that dip to considerably lower values at some frequencies than their nominal rated impedance. In many cases that happens at bass or mid-bass frequencies which typically require a lot of energy. Again, in bridged mode that impedance dip will be seen by the amp as half of its actual value, and the result may be too low a value for some amps to handle with good results.
3)Since in bridged mode both of the channels in the amp participate in the generation of the same output signal, to the extent that the two channels don't have identical characteristics distortion may be introduced in bridged mode that would not occur in stereo mode.
4)In bridged mode a phase inversion has to be applied to the signal on one of the amp's two channels. In some designs that will necessitate insertion of an extra active stage into the signal path, with possible adverse sonic effects.
There are probably other factors that may be involved as well, but those are some that readily come to mind. Despite all that, however, as ZD indicated it is certainly conceivable that there will be some cases in which an amp will sound better in bridged mode than in stereo mode, at least from a subjective standpoint and in the context of some specific systems.